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Math Help - Show arc lengths are equal.

  1. #1
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    Show arc lengths are equal.

    Here's a problem I seen somewhere I thought was kind of fun:

    Show that the arc length of sin(x) over the interval [0,2\pi] is equal to the circumference of the ellipse x^{2}+2y^{2}=2 ".

    The arc length integrals are not easily done by elementary means. At least, I don't think so
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  2. #2
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    Quote Originally Posted by galactus View Post
    Here's a problem I seen somewhere I thought was kind of fun:

    Show that the arc length of sin(x) over the interval [0,2\pi] is equal to the circumference of the ellipse x^{2}+2y^{2}=2 ".

    The arc length integrals are not easily done by elementary means. At least, I don't think so
    There is no exact value of the length of an ellipse, but we can prove the two curves have the same length.
    Attached Thumbnails Attached Thumbnails Show arc lengths are equal.-july34.gif  
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  3. #3
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    Very good curvature. That's how I tackled it. Also, the same way it was done on another forum. That is apparently the best way. I will admit, I scratched my head for a little while before it dawned on me to use parametric arc length. Then , it was obvious. Fell right into place
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    Quote Originally Posted by galactus View Post
    That's how I tackled it. Also, the same way it was done on another forum. That is apparently the best way. I will admit, I scratched my head for a little while before it dawned on me to use parametric arc length. Then , it was obvious. Fell right into place
    An interesting problem and we enjoyed it. Didnt we?
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  5. #5
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    I get the feeling you may think I actually didn't solve the problem, but said I did it that way. Sorry, if it looks that way but I would'nt be that dishonest.

    I was thinking maybe someone would show another way besides the parametric arc length method. I won't post anymore problems that way.
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    Quote Originally Posted by galactus View Post
    I get the feeling you may think I actually didn't solve the problem, but said I did it that way. Sorry, if it looks that way but I would'nt be that dishonest.
    I am sorry you think so. The way I solved the problem is rather normal that many people (including you) can do it that way.
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  7. #7
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    I'm sorry. No ruffled feathers. I just didn't want you to think that.
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